Step by step
The question
Question: write the quadratic equation with roots 2/3 and 3.
roots 23 and 3
1 Sum and product
The sum of the roots: 2/3 plus 3 is 11/3.
sum = 23 + 3 = 113
To add fractions, we use the same bottom, 3: as in the fractions lesson.
23 + 3 = 23 + 93 = 113
The product: 2/3 times 3 is 2.
product = 23 × 3 = 2
2 Write the equation
The equation with roots r and s is x minus r, times x minus s.
Opened up: x squared, minus the sum times x, plus the product.
x² − (sum)x + product = 0
(x − r)(x − s) = x² − (r + s)x + rs
Put the sum and the product in.
x² − (113)x + 2 = 0
Careful!
The sum goes in with a minus in front: minus 11/3.
Careful: not 3x² + 11x + 6 = 0 ✗
Tidy the signs.
x² − 113x + 2 = 0
There are fractions.
We multiply every part by 3, on both sides.
0 times 3 is still 0.
× 3: no fractions
So the equation is 3x² − 11x + 6 = 0.
3x² − 11x + 6 = 0
3 Check
Let's check: put each root in.
x = 2/3 gives 0.
x = 23: 3(23)² − 11(23) + 6 = 0
And x = 3 gives 0 too.
So the equation is right!
x = 3: 3(3)² − 11(3) + 6 = 0
Let's look back.
First: sum and product.
Then: write the equation.
Last: check.
Answer
3x² − 11x + 6 = 0